Valuation of VIX derivatives

Journal of Financial Economics 108 (2013) 367–391

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Journal of Financial Economics journal homepage: www.elsevier.com/locate/jfec

Valuation of VIX derivatives$ Javier Mencı´a a,n, Enrique Sentana b a b

´, 48, E-28014 Madrid, Spain ˜a, Alcala Banco de Espan CEMFI, Casado del Alisal, 5, E-28014 Madrid, Spain

a r t i c l e i n f o

abstract

Article history: Received 17 November 2011 Received in revised form 30 July 2012 Accepted 21 August 2012 Available online 20 December 2012

We conduct an extensive empirical analysis of VIX derivative valuation models before, during, and after the 2008–2009 financial crisis. Since the restrictive mean-reversion and heteroskedasticity features of existing models yield large distortions during the crisis, we propose generalisations with a time-varying central tendency, jumps, and stochastic volatility, analyse their pricing performance, and implications for term structures of VIX futures and volatility ‘‘skews.’’ We find that a process for the log of the observed VIX combining central tendency and stochastic volatility reliably prices VIX derivatives. We also uncover a significant risk premium that shifts the long-run volatility level. & 2012 Elsevier B.V. All rights reserved.

JEL classification: G13 Keywords: Central tendency Stochastic volatility Jumps Term structure Volatility skews

1. Introduction It is now widely accepted that the volatility of financial assets changes stochastically over time, with fairly calmed phases being followed by more turbulent periods of uncertain length. For financial market participants, it is

$ We are grateful to Francis Longstaff (the referee), Dante Amengual, Torben Andersen, Max Bruche, Peter Carr, Rob Engle, Javier Gil-Bazo, Ioannis Paraskevopoulos, Francisco Pen˜aranda, Eduardo Schwartz, Neil Shephard and Dacheng Xiu, as well as seminar participants at Oxford MAN Institute, Universidad de Murcia, Universitat Pompeu Fabra, the SoFiE European Conference (Geneva, June 2009), the World Congress of the Econometric Society (Shanghai, August 2010), the Finance Forum (Elche, November 2010) and the Symposium of the Spanish Economic Association (Madrid, December 2010) for helpful comments, discussions and suggestions. Of course, the usual caveat applies. Financial support from the Spanish Ministry of Science and Innovation through Grants ECO 2008-00280 and 2011-26342 (Sentana) is gratefully acknowledged. The views expressed in this paper are those of the authors, and do not reflect those of the Bank of Spain. n Corresponding author. Tel: þ34 91 338 5414, fax: þ 34 91 338 6102. E-mail addresses: [email protected] (J. Mencı´a), sentana@cemfi.es (E. Sentana).

0304-405X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jfineco.2012.12.003

of the utmost importance to understand the nature of those variations because volatility is a crucial determinant of their investment decisions. Although many model-based and model-free volatility measures have been proposed in the academic literature (see Andersen, Bollerslev, and Diebold, 2009, for a recent survey), the Chicago Board Options Exchange (CBOE) volatility index, widely known by its ticker symbol VIX, has effectively become the standard measure of volatility risk for investors in the US stock market. The goal of the VIX index is to capture the volatility (i.e., standard deviation) of the Standard and Poor’s 500 (S&P500) over the next month implicit in stock index option prices. Formally, it is the square root of the risk-neutral expectation of the integrated variance of the S&P500 over the next 30 calendar days, reported on an annualised basis. Despite this rather technical definition, both financial market participants and the media pay a lot of attention to its movements. To some extent, its popularity is due to the fact that VIX changes are negatively correlated to changes in stock prices. The most popular explanation is that investors trade options on the S&P500 to buy protection in periods of market turmoil, which

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increases the value of the VIX. In fact, as Andersen and Bondarenko (2007) and many others show, the VIX almost uniformly exceeds realised volatility because investors are, on average, willing to pay a sizeable premium to acquire a positive exposure to future equity-index volatility. For that reason, some commentators refer to it as the market’s fear gauge, even though a high value does not necessarily imply negative future returns. But apart from its role as a risk indicator, nowadays it is possible to directly invest in volatility as an asset class by means of VIX derivatives. Specifically, on March 26, 2004, trading in futures on the VIX began on the CBOE Futures Exchange (CFE). They are standard futures contracts on forward 30-day implied vols that cash settle to a special opening quotation (VRO) of the VIX on the Wednesday that is 30 days prior to the 3rd Friday of the calendar month immediately following the expiring month. Further, on February 24, 2006, European-style options on the VIX index were also launched on the CBOE. Like VIX futures, they are cash settled according to the difference between the value of the VIX at expiration and their strike price. Importantly, they can also be interpreted as options on VIX futures, which one can exploit to simplify their valuation and avoid compounding errors made in valuing futures. VIX options and futures are among the most actively traded contracts at CBOE and CFE, averaging close to 260,000 contracts combined per day in 2010, with much larger volumes on certain days.1 One of the main reasons for the high interest in these products is that VIX derivative positions can be used to hedge the risks of investments in the S&P500 index. Szado (2009) finds that such strategies do indeed provide significant protection, especially in downturns. Moreover, by holding VIX derivatives investors can achieve exposure to S&P500 volatility without having to delta hedge their S&P500 option positions with the stock index. As a result, it is often cheaper to be long in out-of-the-money VIX call options than to buy out-of-the-money puts on the S&P500. Due to this possibility, VIX options are the only asset in which open interest is highest for out-of-themoney call strikes (Rhoads, 2011). Although these new assets certainly offer additional investment and hedging opportunities for financial market participants, their correct use requires reliable valuation models that adequately capture the features of the underlying volatility index. In turn, the empirical performance of those valuation models can shed some light on the stochastic process for the VIX, and stock market volatility more generally, which is of interest to academic researchers. Moreover, the prices of volatility derivatives contain valuable information about the views financial market participants hold about the future, and we need

1 For comparison purposes, this value represents about 37% of the average daily volume of S&P500 options, which was 695,000 in 2010. However, the volume of VIX derivatives is growing at a much faster rate. Specifically, the volume of VIX options already represented 52% of the volume of S&P500 options during the first ten months of 2011. In this sense, Rhoads (2011) points out that the volume traded is executed over a smaller time frame (7:20 am–3:15 pm Central Time) than for other derivatives markets.

reliable valuation models to make the correct inferences in a world with risk-averse agents. Somewhat surprisingly, several theoretical approaches to price VIX derivatives appeared in the academic literature long before they could be traded. Specifically, Whaley (1993) priced volatility futures assuming that the observed volatility index on which they are written follows a Geometric Brownian Motion (GBM). As a result, his model does not allow for mean-reversion in the VIX, which, as we shall see, seems to be at odds with the recent empirical evidence.2 The two most prominent meanreverting models proposed so far for volatility indexes have been the square root process (SQR) considered by ¨ Grunbichler and Longstaff (1996), and the log-normal Ornstein-Uhlenbeck (LOU) process analysed by Detemple and Osakwe (2000). Several authors have previously looked at the empirical performance of these pricing models for VIX derivatives. In particular, Zhang and Zhu (2006) study the empirical validity of the SQR model by first estimating its parameters from VIX historical data, and then assessing the pricing errors of VIX futures implied by those estimates. Following a similar estimation strategy, Dotsis, Psychoyios, and Skiadopoulos (2007) also use VIX futures data to evaluate the gains of adding jumps to an SQR diffusion. In addition, they estimate a GBM process. Surprisingly, this model yields reasonably good results, but the time span of their sample is perhaps too short for the mean-reverting features of the VIX to play any crucial role. In turn, Wang and Daigler (2011) compare the empirical fit of the SQR and GBM models using data on options written on the VIX. They also find evidence supporting the GBM assumption. However, one could alternatively interpret their findings as evidence in favour of the LOU process, which also yields the Black (1976) option formula if the underlying instrument is a VIX futures contract, but at the same time is consistent with mean-reversion. Despite this empirical evidence, both the SQR and the LOU processes show some glaring deficiencies in capturing the strong persistence of the VIX, which produces large and lasting deviations of this index from its longrun mean. In contrast, the implicit assumption in those models is that this volatility index mean-reverts at a simple, non-negative exponential rate. Such a limitation becomes particularly apparent during bearish stock markets, in which volatility measures such as the VIX typically experience large increases and remain at high levels for long periods. Arguably, the apparent success of those models is to a large extent due to the fact that the sample periods considered in the existing studies only cover the relatively long and quiet bull market that ended in the summer of 2007.

2 The stationarity of volatility seems to depend on the historical period considered. Schwert (1990) and Pagan and Schwert (1990) find strong evidence for a unit root in stock volatility if the data span the 1930s. In contrast, Schwert (2011) convincingly argues that during the 2008–2009 crisis, volatility exhibits more mean-reversion than in the past.

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

In this context, the initial objective of our paper is to study the empirical ability of existing mean-reverting models for volatility indexes to price derivatives on the VIX over a longer sample span that includes data before, during, and after the unprecedented recent financial crisis, which provides a unique testing ground for our study. To do so, we use an extensive database that comprises all futures and European option prices on the VIX from February 2006, when options were introduced, until December 2010. As a result, we can study whether the SQR and LOU models are able to yield reliable in- and outof-sample prices in a variety of market circumstances. Our findings indicate that although a LOU process for the VIX provides a better fit than an SQR model, especially for VIX options, its performance deteriorates during the market turmoil of the second part of our sample. For that reason, we consider an extension of the SQR model advocated by Bates (2012), among others, which allows for a time-varying central tendency in the mean as well as stochastic volatility that is unspanned by the VIX. Similarly, we generalise the LOU process by proposing several novel but empirically relevant extensions: a time-varying central tendency in the mean, jumps, and stochastic volatility. A central tendency, which was first introduced by Jegadeesh and Pennacchi (1996) and Balduzzi, Das, and Foresi (1998) in the context of term structure models, allows the ‘‘average’’ volatility level to be time-varying, while stochastic volatility permits a changing dispersion for the (log) volatility index, and together with jumps, introduces non-normality in its conditional distribution. Importantly, we study the role that risk premiums play in reconciling the dynamics of the VIX with the prices of VIX derivatives in order to determine the existence of economically important systematic risks. We estimate the SQR and LOU models by maximum likelihood. However, a closed-form expression for the likelihood is not available for many of the extensions. As a consequence, following Trolle and Schwartz (2009), among others, we specify our extensions in state space form and calibrate their parameters by pseudo-maximum likelihood. We use both derivative prices and historical observations on the VIX itself since we want to estimate both real and risk-neutral measures. In order to compute the theoretical derivatives prices, we often need to invert the conditional characteristic function using Fourier methods (see Carr and Madan, 1999; Amengual and Xiu, 2012). To validate the models, we analyse the discrepancies between actual and theoretical derivatives prices. But we also go beyond pricing errors, and analyse the implications of the aforementioned extensions for the term structures of VIX futures, and the option volatility ‘‘skews,’’ all of which are of considerable independent interest. Since we combine futures and options data, we can also assess which of those additional features is more relevant for pricing futures, and which one is more important for options. Importantly, our approach differs from Sepp (2008), who prices VIX derivatives by using data on the S&P500 only. Although the relevant distribution of future values of the VIX conditional on current information might

369

arguably depend not only on the current level of the VIX but also on the value of the S&P500, it seems odd to completely neglect the information content of contemporaneously observed VIX values. This is particularly relevant in view of the fact that one cannot reproduce the spot VIX index by using the model proposed by Sepp (2008) for the purposes of obtaining the required S&P500 option values that feed in the closed-form CBOE formula for this volatility index.3 In order to avoid compounding valuation errors, therefore, we prefer to treat the current VIX level as our ‘‘sufficient statistic,’’ and to treat VIX futures likewise for the purposes of valuing options, instead of assigning that role to the S&P500. The same assumption has recently been made by Song and Xiu (2012). Ideally, though, one would like to use both, but we leave this for future research. In this sense, it is important to emphasise that the univariate stochastic processes that we consider for the VIX are not necessarily incompatible with models for S&P500 options capable of reproducing this observed volatility index. Our analysis can also shed some light on the longlasting debate surrounding the modeling and stationarity of the (instantaneous) volatility of this broad stock market index. In particular, given the relationship between the observed VIX and the unobserved integrated volatility of the S&P500, our results imply that the sophisticated stochastic volatility models that many researchers and practitioners use for valuing S&P options should probably allow for a slower mean-reversion than usually considered, as well as for time-varying volatility of volatility. The rest of the paper is organised as follows. We describe the data in Section 2, and explain our estimation strategy in Section 3. Then, we assess the empirical performance of existing models in Section 4 and consider our proposed extensions in Section 5. Finally, we conclude in Section 6. Auxiliary results are gathered in several Appendices. 2. Preliminary data analysis 2.1. The CBOE volatility index VIX was originally introduced in 1993 to track the BlackScholes implied volatilities of options on the S&P100 with near-the-money strikes (see Whaley, 1993). The CBOE redefined the index in 2003, renamed the original index as VXO, and released a time series of daily closing prices starting in January 1990 (see Carr and Wu, 2006). Nowadays, VIX is computed in real time using as inputs the mid bid–ask market prices for most calls and puts on the S&P500 index for the front month and the second month expirations with at least eight days left (see Chicago Board Options Exchange, 2009).4 Since VIX is expressed in annualised 3 A relevant analogy would be the use of a model for the dividends and risk premia of the 500 individual stocks that constitute the index for the purposes of valuing S&P500 derivatives discarding the information in the index itself, even though any such multivariate model would very likely fail to reproduce the value of the index. 4 Currently, CBOE applies the VIX methodology to three-month options on the S&P500 (VXV), as well as one-month options on the

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pffiffiffiffiffiffiffiffiffi terms, investors typically divide it by 16ð ¼ 256Þ to gauge the expected size of the daily movements in the stock market implied by this index (see Rhoads, 2011). Fig. 1a displays the entire historical evolution of the VIX. Between January 1990 and December 2010, its average closing value was 20.4. As other volatility measures, though, it is characterised by swings from low to high levels, with a temporal pattern that shows meanreversion over the long run but displays strongly persistent deviations from the mean during extended periods. The lowest closing price (9.31) corresponds to December 22, 1993. Fig. 1b, which focuses on the sample period in our derivatives database, shows that volatility was also remarkably low between February 2006 and July 2007, with values well below 20. During this period, the lowest value was 9.89 on January 24, 2007, in what some have called ‘‘the calm before the storm.’’ Indeed, although the Dow Jones Industrial Average closed above 14,000 for the first time in history in the summer of 2007, some warning signals were observed around this period. In particular, on June 22, 2007, Bear Stearns pledged up to $3.2 billion in loans to bail out one of its hedge funds, which was collapsing due to bad bets on subprime mortgages. Moreover, on July 18, 2007, this investment bank disclosed that its two subprime hedge funds had lost all of their value, and one day later Fed Chairman Ben Bernanke warned the US Senate Banking Committee that subprime losses could top $100 billion. Perhaps not surprisingly, over the following year VIX increased to values between 20 and 35. Finally, in the autumn of 2008 it reached unprecedented levels. In particular, the largest historical closing price (80.86) took place on November 20, 2008, although on October 24 the VIX reached an intraday value of 89.53. After this peak, VIX followed a decreasing trend over the following months until the beginning of April 2010, when the Greek debt crisis started worsening. These markedly different regimes offer a very interesting testing ground to analyse the performance of valuation models for volatility derivatives. In particular, we can assess if the models calibrated with pre-crisis data perform well under the extreme conditions of the 2008–2009 financial crisis. In order to characterise the time-series dynamics of the VIX, we have estimated several autoregressive-movingaverage (ARMA) models using the entirety of VIX historical observations from 1990 to 2010; see French, Schwert, and Stambaugh (1987) for a related analysis. Fig. 2a compares the sample autocorrelations of the log VIX with those implied by the estimated models. There is clear evidence of high persistence, with a first-order autocorrelation above 0.98 and a slow rate of decay for higher orders. Consequently, an AR(1) model seems unable to capture the shape of the sample correlogram. An alternative illustration of the failure of this model is provided by the presence of positive partial autocorrelations of orders higher than one, as Fig. 2b confirms. Therefore, it is necessary to introduce a moving

average component to take into account this feature. An ARMA(1,1) model, though, only offers a slight improvement. In contrast, an ARMA(2,1) model turns out to yield autocorrelations and partial autocorrelations that are much closer to the sample values. As we shall see below, our preferred continuous time models have ARMA(2,1) representations in discrete time. Importantly, this result is not due to the behaviour of the VIX during the financial crisis, since an ARMA(2,1) is also necessary to capture the autocorrelation profile when we only consider data from 1990 until the summer of 2007. We have also analysed the presence of time-varying features in the volatility of the log VIX. In particular, we have tested for generalised autoregressive conditional heteroskedasticity (GARCH) effects in the residuals of the ARMA(2,1) estimation.5 Interestingly, we can easily reject the conditional homoskedasticity of the log VIX at

(footnote continued) most important US stock market indexes: DJIA (VXD), S&P100 (VXO), Nasdaq-100 (VXN), and Russell 2000 (RVX). They also construct analogous short-term volatility indexes for Crude Oil (OVX), Gold (GVZ), Soybean (SIV), Corn (CIV), and the US $=h exchange rate (EVZ).

5 Following Demos and Sentana (1998), we carry out a one-sided Lagrange multiplier (LM) test of conditional homoskedasticity against 2 GARCH as T R~ from the regression of the squared ARMA(2,1) residuals on a constant and the RiskMetrics volatility estimate (see RiskMetrics Group, 1996).

80 70 60 50 40 30 20 10 May90

Jan93

Oct95

Jul98

Apr01

Jan04

Oct06

Jul09

90 20−Nov−08

80 70 60

21−Jun−07 15−Aug−08

50

31−Mar−10

40 30 20 10 Mar06 Oct06

Apr07

Nov07 Jun08 Dec08

Jul09

Jan10 Aug10

Fig. 1. Historical evolution of the VIX index. (a) 1990–2010. (b) 2006–2010.

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391 1 AR(1) ARMA(1,1) ARMA(2,1) Actual

0.98 0.96 0.94 0.92 0.9 0.88 0.86 1

2

3

4

5

6

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0.12 AR(1) ARMA(1,1) ARMA(2,1) Actual

0.1 0.08 0.06 0.04 0.02 0 −0.02 2

3

4

5

6

7

8

9

10

Fig. 2. Time series autocorrelations of the log-VIX and estimated ARMA models. (a) Autocorrelations. (b) Partial autocorrelations. Results are based on the 1990–2010 sample (5,280 daily observations).

all conventional levels. As we shall in Section 5.4, the volatility of our preferred continuous time specification closely matches the volatility of an ARMA(2,1)-GARCH(1,1) model. 2.2. VIX derivatives Our sample contains daily closing bid–ask mid prices of futures and European put and call options on the VIX, which we downloaded from Bloomberg. We apply several filters to ensure the reliability of the data that we finally use. Following Dumas, Fleming, and Whaley (1998), we exclude derivatives with fewer than six days to expiration due to their illiquidity. In addition, we only retain those prices for which open interests and volumes are available. We consider the entire history of these series since options were introduced in February 2006 until December 2010. In terms of maturity, we have data on all the contracts with expirations between March 2006 and May (July) 2011 for options (futures). CFE may list futures for up to nine near-term serial months, as well as five months on the February quarterly cycle associated to the March quarterly cycle for options on the S&P500. In turn,

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CBOE initially lists some in-, at-, and out-of-the-money strike prices, and then adds new strikes as the VIX index moves up or down. Generally, the options expiration dates are up to three near-term months plus up to three additional months on the February quarterly cycle.6 All in all, we have 8,665 and 87,870 prices of futures and options, respectively. Of those option prices, 58,099 correspond to calls and 29,771 to puts. We have between four and six daily futures prices during 2006; afterwards, eight prices per day become available, on average. The number of option prices per day is also smaller at the beginning of the sample, but it tends to stabilise in 2007 at around 20 for puts and 40 for calls. The number of call prices per day increases again after mid-2009 and is greater than 60 at the end of the sample. We proxy for the riskless interest rate by using the daily Eurodollar rates at one-week, one, three and six months, and one year, which we interpolate to match the maturities of the futures and option contracts that we observe.7 Fig. 3a shows the evolution of the VIX term structure implicit in VIX futures. Futures prices remained low albeit above the volatility index until mid-2007, which suggests that market participants perceived that the VIX was too low during those years. For instance, on October 12, 2006, the VIX was 11.09, while the price of VIX futures expiring on February 14, 2007, was 15.72. The spot price reflected the expected volatility for the period of October 12– November 11, while the futures price reflected the expected volatility for the period of February 14–March 16, 2007. Fig. 3a clearly confirms that there was an initial increase in the summer of 2007 and a substantial level shift in the last quarter of 2008. The negative slope during the latter period, though, indicates that the market did not expect the VIX to remain at such high values forever, a fact already highlighted by Schwert (2011). Eventually, this prediction turned out to be on the winning side and futures prices significantly came down until April 2010, right before the beginning of the European sovereign debt crisis. Figs. 3b and c show the average implied volatility skews implicit in VIX option prices for different times to maturity. We consider low volatility days in Fig. 3b and high volatility days in Fig. 3c. We define low (high) volatility days as those in which the implied volatility of one-month at-the-money options remains below (above) their average value over the sample. Interestingly, we observe a consistent pattern of implied volatility smirks with positive slope regardless of the time to maturity and volatility of the VIX. There only seems to be a small change in slope for deep in-the-money calls (or out-of-the-

6 VIX futures contracts have a multiplier of 1,000, while VIX option contracts have a multiplier of 100. This means that the value of the VIX futures contract is determined by multiplying $1,000 times the quoted futures price. On March 2, 2009, the CBOE introduced mini-VIX futures, which have a multiplier of 100 (see Rhoads, 2011). 7 Given that Eurodollar and London interbank offered rates (LIBOR) are very similar, the results are unlikely to be sensitive to this choice. As usual in the option pricing literature, we assume constant interest rates in our models. Although interest rates are of course not constant in practice, we have checked that their changes are not significantly correlated with changes in the VIX.

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(November 2008) because the market expected a fall in the volatility of the VIX. 3. Pricing and estimation strategy

60 40 20 8 Mo nth

6 s to

4

exp

irat

ion

2

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Nov1 0

Date

0.45 0.4 0.35 0.3 0.25 15 ≤τ<45 45 ≤τ<75 75 ≤τ<105 105 ≤τ<135

0.2 0.15

−0.4

−0.2

0

0.2

0.4

We assume that there is a risk-free asset with instantaneous rate r. Let V(t) be the VIX value at time t. We define Fðt,TÞ as the actual price of a futures contract on V(t) that matures at T 4 t. Similarly, we will denote the prices at t of call and put options maturing at T with strike price K by cðt,T,KÞ and pðt,T,KÞ, respectively. Importantly, since the VIX index is a risk-neutral volatility forecast, not a directly traded asset, there is no cost of carry relationship between the price of the futures and the VIX (see ¨ Grunbichler and Longstaff, 1996, for more details). There is no convenience yield either, as in the case of futures on commodities. Therefore, absent any other market information, VIX derivatives must be priced according to some model for the risk-neutral evolution of the VIX. This situation is similar, but not identical, to term structure models. Let M index the asset pricing models that we consider. Then, the theoretical futures price implied by model M will be: F M ðt,T,VðtÞ, /Þ ¼ EQM ½VðTÞ9IðtÞ, /,

0.6

Moneyness 0.45 0.4

ð1Þ

where Q indicates that the expectation is evaluated at the risk-neutral measure, / is the vector of free parameters of model M, and I(t) denotes the information available at time t, which includes V(t) and its past values. We can analogously express the theoretical value of a European call option with strike K and maturity at T under this model as cM ½t,T,K,Fðt,TÞ, / ¼ expðr tÞEQM ½maxðVðTÞK,0Þ9IðtÞ, /,

0.35

ð2Þ

0.3 0.25 15 ≤τ <45 45 ≤τ <75 75 ≤τ <105 105≤τ <135

0.2 0.15 −0.4

−0.2

0

0.2

0.4

0.6

Moneyness Fig. 3. Term structure of VIX derivatives. (a) Term structure of VIX futures. (b) Implied volatility smirk on low volatility days. (c) Implied volatility smirk on high volatility’days. The lines in Panels B and C show the average implied volatility for a certain moneyness and time to maturity (denoted by t) within a given interval. Implied volatilities have been obtained by inverting the Black (1976) call price formula. Moneyness is defined as logðK=Fðt,TÞÞ. Low (high) volatility days are those in which one-month at-the-money implied volatilities are below (above) their average value over the sample.

where t ¼ Tt. Nevertheless, we can exploit the fact that VðTÞ ¼ FðT,TÞ to price calls using the futures contract that expires on the same date as the underlying instrument, instead of the actual volatility index. In this way, we make sure that the pricing errors of options are not caused by distortions in our futures valuation formulas. Similarly, European put prices pðt,T,KÞ under M can be easily obtained from the put-call-forward parity relationship pM ½t,T,K,Fðt,TÞ, / ¼ cM ½t,T,K,Fðt,TÞ, /expðr tÞ½Fðt,TÞK:

We use futures and options prices, as well as data on the VIX index, which allows us to estimate the risk premia implicit in the market of VIX derivatives. We estimate the parameters by maximum likelihood for the models whose likelihood is known in closed form, and by pseudo-maximum likelihood for the remaining ones. In both cases, we assume the existence of pricing errors such that Fðt,TÞ ¼ F M ðt,T,VðtÞ, /Þ þ xft þ eft,T ,

money puts). Therefore, any pricing model must adequately capture these empirical features. Finally, although Figs. 3b and c show that option prices were higher for longer maturities on average, this is not necessarily the case on all the days in the sample. In particular, option prices decreased with maturity at the peak of the financial crisis

cðt,T,KÞ ¼ cM ½t,T,K,Fðt,TÞ, / þ $ðMðt,TÞÞ½ð1Zo Þxot þ Zo ect,T ðKÞ, pðt,T,KÞ ¼ pM ½t,T,K,Fðt,TÞ, / þ $ðMðt,TÞÞ½ð1Zo Þxot þ Zo ept,T ðKÞ,

where eft,T  Nð0, s2f e Þ, ect,T ðKÞ  Nð0,1Þ, and ept,T ðKÞ  Nð0,1Þ are orthogonal and independent and identically distributed (iid) over time and across strikes and maturities, while

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

xft  Nð0, s2f x Þ and xot  Nð0,1Þ are also orthogonal and iid over time but common for all futures and options, respectively. Thus, we decompose pricing errors into common and idiosyncratic terms. Thanks to this parametrisation, we allow for contemporaneous correlation between the pricing errors. As noted by Bates (2000), ignoring this feature would affect the relative weighting that the options and futures would receive in the estimation. In contrast, the inclusion of common factors avoids overestimating the amount of truly independent information in the data.8 In addition, we parametrise the variances of option pricing errors by means of the quadratic function

$2 ðMðt,TÞÞ ¼ sa þ sb ðMðt,TÞsc Þ2 ,

ð3Þ

where Mðt,TÞ ¼ logðK=Fðt,TÞÞ measures the moneyness of the option. In this way, we avoid giving undue weight to higher priced or in-the-money options whose pricing errors are higher simply because of their larger scale. Importantly, we also employ different parameters for the pricing error variances of futures and options, thereby implicitly adjusting their relative weight in the likelihood. We can write the log likelihood at a particular time t as l½ot ,f t ,VðtÞ9Iðt1Þ ¼ lo ðot 9f t ,VðtÞ,Iðt1ÞÞ þ lf ðf t 9VðtÞ,Iðt1ÞÞ þ lv ½VðtÞ9Iðt1Þ, where ot and f t are, respectively, the set of option and futures prices traded at t, lo ðot 9f t ,VðtÞ,Iðt1ÞÞ is the log density of the option prices conditional on futures prices and VIX, lf ðf t 9VðtÞ,Iðt1ÞÞ is the log likelihood of the futures prices conditional on the volatility index, and lv ½VðtÞ9Iðt1Þ is the log density of the volatility index.9 Thus, the log likelihood reflects that we price futures conditional on the value of the VIX, in the same way as we price options conditional on the value of the futures. In addition, we take into account the contribution of the VIX to the log likelihood in order to obtain the parameters under the real measure. A nontrivial advantage of our estimation method over traditional calibration procedures is that we do not have to treat the model parameters as deterministic functions of time. In addition, we can also obtain standard errors for our parameter estimators, which can in turn be used to conduct formal hypothesis tests. Furthermore, we can exploit the relationship between actual and risk-neutral measures to estimate prices of risk (see Appendices B and C for details). We consider two estimation samples that correspond to two distinct volatility phases described in Section 2: until 15-Aug-2008 and full sample. We use the first sample to evaluate the out-of-sample empirical fit of the models.10 Thus, we can assess model performance following a major volatility increase in global stock markets. 8

Nevertheless, empirical results without the common factors xft and xot , which are available from the authors upon request, yield qualitatively similar results. 9 The usefulness of this additive decomposition is limited, though, because most model parameters affect more than one component. 10 We analyse out-of-sample performance using the parameter estimates obtained in-sample. We also use the current value of the VIX to compute futures prices and the current value of futures prices to

373

4. Existing one-factor models 4.1. Model specification We first compare the two mean-reverting volatility models that have been used so far in the literature: the square root and the log Ornstein-Uhlenbeck processes. As ¨ we mentioned before, Grunbichler and Longstaff (1996) proposed the square root process (SQR) to model a standard deviation index. This model, which was used by Cox, Ingersoll, and Ross (1985) for interest rates and Heston (1993) for the instantaneous variance of stock prices, satisfies the diffusion pffiffiffiffiffiffiffiffiffi P P dVðtÞ ¼ kP ½y VðtÞ dt þ s VðtÞ dW ðtÞ, where W P ðtÞ is a Brownian motion under the real measure. Following Dai and Singleton (2000), we specify a price of risk that is proportional to the instantaneous pffiffiffiffiffiffiffiffiffi volatility. Specifically, we assume that L ðtÞ ¼ B VðtÞ, so that v pffiffiffiffiffiffiffiffiffi Q P dW ðtÞ ¼ dW ðtÞ þ B VðtÞ dt. Then, it is straightforward to show that V(t) satisfies the square root diffusion pffiffiffiffiffiffiffiffiffi Q dVðtÞ ¼ k½y VðtÞ dt þ s VðtÞ dW ðtÞ under the equivalent risk-neutral measure, where k ¼ kP þ P sB and y ¼ kP y =k. As is well known, the risk-neutral distribution of 2cVðTÞ given V(t) is a non-central chi-square with n ¼ 4ky =s2 degrees of freedom and non-centrality parameter c ¼ 2cVðtÞexpðktÞ, where c¼

2k : s2 ð1expðktÞÞ

As a result, the price of the futures contract (1) can be expressed in this case as F SQR ðt,T,VðtÞ, k, y , sÞ ¼ y þ expðktÞ½VðtÞy :

ð4Þ

We can interpret y as the long-run mean of V(t), since the conditional expected value of the volatility index converges to y as t goes to infinity. In addition, k is usually interpreted as a mean-reversion parameter because the higher it is, the more quickly the process reverts to its long-run mean.11 Interestingly, the SQR model introduces stochastic volatility because the conditional variance of the VIX implied by this model is an affine function of V(t) (see Appendix D). The call price formula (2) for this model becomes cSQR ðt,T,K, k, y , sÞ ¼ VðtÞexpððk þ rÞtÞ½1F NC2 ð2cK; n þ 4, cÞ þ y ½1expðktÞexpðr tÞ ½1F NC2 ð2cK; n þ2, cÞ K expðr tÞ½1F NC2 ð2cK; n, cÞ,

ð5Þ

(footnote continued) compute option prices, just as a financial market participant would do in real time. In addition, in the models with more than one factor, we follow the standard procedure of filtering the additional factor using the past evolution of the VIX and its derivatives with the in-sample estimates. 11 As remarked by Hansen and Scheinkman (2009), the density of this distribution will be positive at VðTÞ ¼ 0 if the Feller condition 2ky Z s2 is violated.

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374

where F NC2 ð; n, cÞ is the cumulative distribution function (CDF) of a non-central chi-square distribution with n degrees of freedom and non-centrality parameter c. Hence, it is straightforward to express the call price as a function of Fðt,TÞ by exploiting the relationship between futures prices and V(t) in (4). Subsequently, Detemple and Osakwe (2000) considered the log-normal Ornstein-Uhlenbeck (LOU) diffusion: P

Table 1 Root mean square pricing errors in existing one-factor models. The pricing errors of futures and options on the VIX have been considered. Estimation of the parameters by maximum likelihood using data on the VIX index and its options and futures. ‘‘SQR’’ denotes square root model while ‘‘LOU’’ refers to the log-normal Ornstein-Uhlenbeck process. Aug08–Mar10 estimates

P

d log VðtÞ ¼ k½y log VðtÞ dt þ s dW ðtÞ: If once again we follow Dai and Singleton (2000) in specifying the price of risk as Lv ðtÞ ¼ B, then V(t) will satisfy the LOU diffusion

SQR LOU

Full sample estimates

Feb06–Aug08 In-sample

Aug08–Dec10 Out-of-sample

Feb06–Dec10

0.807 0.779

2.037 1.982

2.454 1.533

Q

d log VðtÞ ¼ k½y log VðtÞ dt þ s dW ðtÞ P

under the risk-neutral measure, where y ¼ y sB=k. As is well known, this model implies that log VðtÞ would follow a conditionally homoskedastic Gaussian AR(1) process if sampled at equally spaced discrete intervals. More generally, the conditional risk-neutral distribution of log VðTÞ given V(t) would be Gaussian with mean

mðt, tÞ ¼ y þ expðktÞ½log VðtÞy  and variance

j2 ðtÞ ¼

s2 2k

½1expð2ktÞ:

ð6Þ

As in the SQR process, y and k can be interpreted as the long-run mean and mean-reversion parameters, respectively, but now it is the log of V(t) that mean-reverts to y . And although (6) shows that the LOU model is homoskedastic for the log VIX, it can be shown that the process followed by the VIX is heteroskedastic due to Jensen’s inequality (see Appendix D for more details). In this context, it is straightforward to show that the futures price is F LOU ðt,T,VðtÞ, k, y , sÞ ¼ expðmðt, tÞ þ 0:5j2 ðtÞÞ, while the call price can be expressed as " # logðFðt,TÞ=KÞ þ 12 j2 ðtÞ cLOU ðt,T,K, k, y , sÞ ¼ expðr tÞFðt,TÞF jðtÞ K expðr tÞF

" # logðFðt,TÞ=KÞ 12 j2 ðtÞ jðtÞ ð7Þ

if we take the futures contract as the underlying instrument, where FðÞ denotes the standard normal CDF. This is the well-known Black (1976) formula, although in this case the implied volatility jðtÞ in (6) is not constant across maturities. In this sense, the pricing formula proposed by Whaley (1993) based on a geometric Brownian motion can also be expressed as (7) if jðtÞ is taken as a constant irrespective of t. 4.2. Empirical performance We have estimated the parameters of these two models over the two sample periods described at the end of Section 3. Table 1 reports the in- and out-of-sample

root mean square pricing errors (RMSE). We observe that the SQR model yields larger in-sample price distortions than the LOU model. Those distortions remain in the out-of-sample period. We have re-estimated the two models with VIX and futures data only to check that the use of option data in the estimation of the parameters is not driving the results. We have found that the LOU model still yields a better performance in that case. Table 2 reports the parameter estimates that we obtain under the real measure, as well as the parameters of the pricing error variances. Although the values are not directly comparable because volatility is expressed in levels in the SQR model and in logs in the LOU model, in both cases the mean-reversion parameter k is quite sensitive to the sample period used for estimation purposes (see footnote 2 and references therein). In contrast, the volatility parameter s is more stable for the LOU process, probably because the log transformation is more appropriate to capture the distortions produced by the large movements of the VIX that took place at the end of our sample. Table 2 also shows that the parameters of the quadratic function (3) are highly significant. Therefore, allowing for different variances of pricing errors across strikes seems to be relevant in ensuring that higher priced options do not receive undue weight. An alternative, more illustrative way to assess the validity of the LOU model is by considering the implied volatilities obtained with the Black (1976) formula (7). As we mentioned before, the implied volatility of the LOU process is constant for different degrees of moneyness, but not across different maturities because (6) depends on t. Hence, if this model were correct, then we should obtain constant implied volatilities for a given maturity regardless of moneyness. As we have seen in Figs. 3b and c, this is not at all the case. In fact, we generally observe that implied vols have a positive moneyness slope. In addition, the comparison of these figures suggests that the volatility of the log VIX is not constant over time, as the LOU model assumes. In this sense, the LOU process is not only unable to generate the observed volatility skews, but it also fails to capture the average level of implied volatilities. As we will see in Section 5.5, though, the SQR model generates volatility skews, but they have the opposite slope to the actual skews in Figs. 3b and c.

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375

Table 2 Parameter estimates of the existing one-factor models. Estimation by maximum likelihood using data on the VIX index and its options and futures from February 06 until December 10. ‘‘SQR’’ denotes square root model while ‘‘LOU’’ refers to the log-normal Ornstein-Uhlenbeck process. Standard errors, displayed in parentheses, have been obtained by using the outer-product of the score to estimate the information matrix. sf e and sf x are the parameters of the futures pricing errors. Zo , sa , sb , and sc are the parameters of the option pricing errors, whose variance is the quadratic function of moneyness (3). Real measure parameters

Pricing error parameters

Model

Estimation

k

y

s

sf e

sf x

sa

Zo

sb

sc

SQR

Full sample

2.555 (0.071) 1.495 (0.015)

24.350 (0.661) 19.666 (0.150)

4.504 (0.005) 2.916 (0.006)

0.122 (0.001) 0.040 (4.1  10  4)

0.081 (0.009) 0.008 (0.001)

0.006 (2.8  10  4) 0.006 (2.6  10  4)

0.497 (0.019) 0.369 (0.024)

0.437 (0.017) 0.275 (0.009)

 0.901 (0.002)  0.727 (0.003)

2.898 (0.004) 1.732 (0.009)

3.099 (0.112) 2.927 (0.171)

0.884 (4.4  10  4) 0.710 (0.001)

0.075 (0.001) 0.042 (4.2  10  4)

0.030 (0.003) 0.008 (0.001)

0.006 (2.4  10  4) 0.005 (2.2  10  4)

0.455 (0.017) 0.412 (0.021)

0.207 (0.007) 0.174 (0.006)

 0.901 (0.002)  0.773 (0.005)

Until Aug 08 LOU

Full sample Until Aug 08

and y .12 More explicitly,

5. Extensions

EQ ½VðTÞ9IðtÞ ¼ y þ dðtÞ½yðtÞy  þ expðktÞ½VðtÞyðtÞ,

5.1. Model specification

ð10Þ

where The results of the previous section indicate that a LOU process offers a better empirical fit than a SQR process. Unfortunately, their performance tends to deteriorate during the recent financial crisis. For that reason, in this section we explore several extensions to those models. For the sake of brevity, though, we focus on the riskneutral measure and the prices of risk. Further details can be found in Appendices B and C. Specifically, we extend the SQR model by considering the concatenated SQR (CSQR) proposed by Bates (2012), among others: pffiffiffiffiffiffiffiffiffi Q dVðtÞ ¼ k½yðtÞVðtÞ dt þ s VðtÞ dW v ðtÞ, with dyðtÞ ¼ k ½y yðtÞ dt þ s

pffiffiffiffiffiffiffiffi yðtÞ dW Qy ðtÞ:

W Qv ðtÞ

Q

k k expðktÞ expðktÞ: kk kk

ð11Þ

Importantly, the CSQR model introduces stochastic volatility that is not spanned by the VIX, since the conditional variance under this model is an affine function of V(t) and yðtÞ (see Appendix B). In turn, we add three empirically relevant features to the LOU model: a time-varying mean, jumps, and stochastic volatility. We consider these extensions first in isolation, and then in combination. As a general rule, we model the price of risk of the continuous part of the diffusions as in the onefactor models, while we assume that jump risk is not priced. All in all, we compare the following cases:  Central tendency (CTOU):

We specify the prices of risk that link the Wiener processes under P and Q as pffiffiffiffiffiffiffiffiffi Q P dW v ðtÞ ¼ dW v ðtÞ þ B VðtÞ dt, ð8Þ pffiffiffiffiffiffiffiffi Q P dW y ðtÞ ¼ dW y ðtÞ þ B yðtÞ dt,

dðtÞ ¼

ð9Þ

where and W y ðtÞ are independent Brownian motions. These specifications ensure that the VIX follows the same process under the real measure. As we mentioned before, an important deficiency of previously existing models is that they assume that the volatility index mean-reverts at a simple, non-negative exponential rate, which is in fact zero in the GBM model proposed by Whaley (1993). However, the long, persistent swings in the VIX in Fig. 1a suggest that we need to allow for more complex dynamics. In this sense, the CSQR allows the VIX to revert towards a central tendency, which in turn fluctuates stochastically over time around a longrun mean y . As a consequence, the conditional mean of VðTÞ is a function of the distance between V(t) and the central tendency yðtÞ, as well as the distance between yðtÞ

Q

d log VðtÞ ¼ k½yðtÞlog VðtÞ dt þ s dW v ðtÞ, with Q

dyðtÞ ¼ k ½y yðtÞ dt þ s dW y ðtÞ, W Qv ðtÞ

ð12Þ

Q

and W y ðtÞ are independent Brownian where motions. We specify the prices of risk that link the Wiener processes under P and Q as Q

P

Q

P

dW v ðtÞ ¼ dW v ðtÞ þ B dt, dW y ðtÞ ¼ dW y ðtÞ þ B dt:

ð13Þ ð14Þ

These specifications ensure that the process followed by the log VIX is also affine under the real measure. We show in Appendix C that the exact discretisation of log VðtÞ in the above model is a Gaussian ARMA(2,1) process, which is consistent with the evidence reported in Section 2 (see Fig. 2). Therefore, its likelihood 12 Unlike in the SQR model, the convergence of (10) to the long-run mean is not necessarily a monotonic function of t.

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

376

function can be computed in closed form. As discussed in Jegadeesh and Pennacchi (1996) and Balduzzi, Das, and Foresi (1998), (12) allows the volatility index to revert towards a time-varying central tendency whose long-run mean is y . As a consequence, the conditional mean of log VðTÞ is a function of the distance between log VðtÞ and the central tendency yðtÞ, as well as the distance between yðtÞ and y .13 More explicitly, EQ ½log VðTÞ9IðtÞ ¼ y þ dðtÞ½yðtÞy  þexpðktÞ½log VðtÞyðtÞ,

ð15Þ

where dðtÞ is given by (11). Notice that, although (10) and (15) have a similar structure, the two formulas are not equivalent because the VIX appears in levels in (10). In addition, the CTOU model can equivalently be expressed as the ‘‘superposition’’ (i.e., sum) of two LOU factors. Therefore, the structure in (12) is not restrictive for the CTOU.14  Jumps (LOUJ): d log VðtÞ ¼ k½y log VðtÞ dt þ s

Q dW v ðtÞ þ dZðtÞ

l dt, kd ð16Þ W Qv ðtÞ,

where Z(t) is a pure jump process independent of with intensity l, and whose jump amplitudes are exponentially distributed with mean 1=d, or ExpðdÞ for short. Note that the last term in (16) simply introduces a constant shift in the distribution of log VðtÞ which ensures that y remains the long-run mean of log VðtÞ. Jumps in models for instantaneous volatility have been previously considered by Duffie, Pan, and Singleton (2000) and Eraker, Johannes, and Polson (2003), among others, while Todorov and Tauchen (2011) also consider jumps in modeling the VIX. Unlike pure diffusions, this model allows for sudden movements in volatility indexes, which nevertheless have lasting effects due to the fact that the mean-reversion parameter k is bounded. Again, we assume (13) for the price of diffusion risk.  Stochastic volatility (LOUSV): pffiffiffiffiffiffiffiffiffiffi Q d log VðtÞ ¼ k½y log VðtÞ dt þ oðtÞ dW v ðtÞ, where oðtÞ follows an OU-G process, which belongs to the class of Le´vy OU processes considered by BarndorffNielsen and Shephard (2001). Specifically, doðtÞ ¼ l oðtÞ dt þ dZ ðtÞ,

ð17Þ

where Z ðtÞ is a pure jump process with intensity l and Expðd Þ jump amplitude, while W Qv ðtÞ is an independent Brownian motion. We use this extension to assess to what extent the price distortions in the previous models are due to the assumption of constant volatility over time. Importantly, the model that we adopt is consistent with the presence of mean-reversion in oðtÞ, since EQ ½oðTÞÞ9oðtÞ ¼ d

1

þ expðl tÞ½oðtÞd

1

:

ð18Þ

13 As in the CSQR model, the convergence of (15) to the long-run mean is not necessarily a monotonic function of t. 14 The ‘‘component’’ GARCH model proposed by Ding and Granger (1996) to capture ‘‘long memory’’ features of volatility (see also Engle and Lee, 1999) can be regarded as a discrete time analogue to (12).

1

Hence, d can be interpreted as the long-run mean of the instantaneous volatility of the log VIX, while l will be the corresponding mean-reversion parameter. Another nontrivial advantage of this model over other alternatives such as a square root process for oðtÞ is that it allows the valuation of derivatives by inverting the conditional characteristic function (see Appendix C for details). Once again, we consider a price of diffusion risk proportional to instantaneous volatility to ensure that the log VIX under the real measure remains an affine process (see Dai and Singleton, 2000). Specifically, we assume pffiffiffiffiffiffiffiffiffiffi Q P dW v ðtÞ ¼ dW v ðtÞ þ Bo oðtÞ dt: ð19Þ Under this specification, the price of risk has a stronger impact when the volatility of the VIX is larger.  Central tendency and jumps (CTOUJ): Q

d log VðtÞ ¼ k½yðtÞlog VðtÞ dt þ s dW v ðtÞ þ dZðtÞ

l

kd

dt,

ð20Þ where yðtÞ follows the diffusion (12) and Z(t) is a pure jump process with intensity l and ExpðdÞ jump amplitude, while W Qv ðtÞ and W Qy ðtÞ are independent Brownian motions. We again introduce a constant shift in (20) to ensure that y is the long-run mean of both yðtÞ and log VðtÞ. As in the CTOU model, we assume that the prices of risk are given by (13) and (14).  Central tendency and stochastic volatility (CTOUSV): pffiffiffiffiffiffiffiffiffiffi Q d log VðtÞ ¼ k½yðtÞlog VðtÞ dt þ oðtÞ dW v ðtÞ, where yðtÞ follows the diffusion (12) while oðtÞ is defined in (17). As in previous cases, the jump variable Z ðtÞ and the Brownian motions are mutually independent. Similarly, we consider the prices of risk specifications (14) and (19). Despite the apparent differences between (13), (14), and (19), they are all special cases of a generic specification. In particular, following Cheridito, Filipovic´, and Kimmel (2007), we can write the risk-neutral probability measure in terms of the real measure for all the extensions that we consider as  Z T q ¼ exp  ðLv ðsÞdW v ðsÞ þ Ly ðsÞdW y ðsÞÞ t

1  2

Z

T t

 ðL2v ðsÞ þ L2y ðsÞÞ ds ,

pffiffiffiffiffiffiffiffiffiffi where Lv ðtÞ ¼ Bo oðtÞ and Ly ðtÞ ¼ B in our case.15 Except for the CTOU model, it is not generally possible to price derivative contracts for these extensions in closed form. However, it is possible to obtain the required prices by Fourier inversion of the conditional characteristic function. In particular, we use formula (5) in Carr and Madan (1999) to invert the relevant characteristic functions of the extensions to the LOU model (see Appendix C). However, this approach is not valid for the CSQR 15 Note that oðtÞ ¼ s2 in the extensions without stochastic volatility. Accordingly, we define B ¼ Bo s in those cases.

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

model, where we need to follow recent results by Amengual and Xiu (2012) to invert the characteristic function (see Appendix B). Some of the previous extensions introduce as additional factors a time-varying tendency, a time-varying volatility, or both, which we need to filter out for estimation purposes. In this regard, we use the standard Kalman filter to ‘‘estimate’’ yðtÞ. And although the jump variable Z(t) in (16) can also be interpreted as an additional factor, its impact cannot be separately identified from the impact of Q the diffusion shocks dW v ðtÞ because both variables share the same mean-reversion coefficient k. Finally, following Trolle and Schwartz (2009) and others, we employ the extended Kalman filter to deal with oðtÞ (see once again Appendix C for further details).

377

this model is able to describe the out-of-sample behaviour of the VIX during the more extreme periods of the financial crisis. Fig. 4 compares the empirical CDFs of the square pricing errors of futures and option prices separately. This figure shows that the CTOUSV model dominates in the first-order stochastic sense all the other models for both futures and options. The ordering of the remaining models, though, depends on the type of derivative asset considered. In the case of futures, the CTOUSV is closely followed by the CSQR and then by the other central tendency specifications (CTOU and CTOUJ), which display almost identical results. In turn, they are followed by the 1 0.9

5.2. Empirical performance

0.8

Table 3 reports the in- and out-of-sample RMSEs of the extensions introduced in the previous section. As expected, the CSQR process is able to yield much smaller RMSEs than the SQR and LOU models, both in- and out-ofsample. Nevertheless, the CTOU model achieves a slightly superior fit, especially out-of-sample. In any case, a persistent time-varying mean seems to capture a crucial feature of the data. On the other hand, the introduction of jumps in the LOU model does not introduce improvements in the aggregate RMSEs. Similarly, adding jumps to the CTOU model does not substantially improve the fit either. However, it is important to emphasise that Table 3 does not assess the importance of jumps on the historical dynamics of the VIX, only their pricing implications. In contrast, we find significant improvements when we consider stochastic volatility. The LOUSV introduces important reductions in the RMSEs, but the combination of central tendency and stochastic volatility provided by the CTOUSV model yields the overall best fit. Importantly,

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Square pricing error 1 0.9 0.8 0.7 0.6

Table 3 Root mean square pricing errors in the extended models. The pricing errors of futures and options on the VIX have been considered. Estimation of the parameters by maximum likelihood for the CTOU model, and by pseudo-maximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures. ‘‘LOUJ’’ introduces jumps in the log-normal OrnsteinUhlenbeck process (LOU), whose size follows an exponential distribution. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated square root process. Aug08-Mar10 estimates

CTOU LOUJ LOUSV CTOUJ CTOUSV CSQR

Full sample estimates

Feb06–Aug08 In-sample

Aug08–Dec10 Out-of-sample

Feb06–Dec10

0.348 0.837 0.664 0.354 0.232 0.424

0.713 2.160 1.617 0.691 0.344 1.156

0.655 1.634 1.303 0.631 0.306 0.691

0.5

SQR LOU CTOU LOUJ LOUSV CTOUJ CTOUSV CSQR

0.4 0.3 0.2 0.1 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Square pricing error Fig. 4. Empirical cumulative distribution function of the square pricing errors. (a) Futures. (b) Options. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘LOUJ’’ introduces jumps in the LOU model, whose size follows an exponential distribution. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated SQR model. Estimation by maximum likelihood for the SQR, LOU and CTOU models, and by pseudomaximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10.

378

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LOUSV, LOUJ, LOU, and SQR models. But for options, the second-best model is LOUSV, which is followed by the CTOUJ, LOUJ, CTOU, LOU, SQR, and CSQR models. Thus, we observe that the central tendency is relatively more important for futures, while stochastic volatility offers greater gains on options. At the same time, a central tendency does not harm the option pricing performance of the CTOUSV model, while stochastic volatility does not cause any distortions to futures prices. This is not surprising, given that we show in Appendix C that option prices for the extensions of the LOU model do not depend on yðtÞ once we condition on the current futures price. As for jumps, Fig. 4 shows that they do indeed help in pricing options but they hardly provide any improvement for futures. The small impact of jumps on futures is again to be expected because we can also show that jumps in the LOUJ model generate identical futures prices as a LOU model up to first order.16 And, compared to stochastic volatility, jumps seem to yield a minor improvement even for options. In this sense, Bakshi, Cao, and Chen (1997) find that, once stochastic volatility is modeled, adding jumps only leads to second-order pricing improvements. Lastly, the SQR model offers the worst overall results, while the CSQR model also yields a poor fit for option prices. Fig. 5a compares the ‘‘actual’’ futures prices for a constant 30-day maturity, which we obtain by interpolation of the adjacent contracts, with the daily estimates of 30-day futures prices generated by the SQR, LOU, and CTOU models. By focusing on a constant maturity, we can not only compare the absolute magnitude of the pricing errors, but also their sign and persistence. As can be seen, the pricing errors of the SQR process are larger than those of the LOU process, especially after mid-2007. In turn, those of the LOU model are not only substantially larger than those of the CTOU, but they also display much stronger persistence. For instance, the SQR and LOU models systematically underprice futures from October 2008 until the end of the sample. The slower meanreverting properties of the VIX are probably responsible for these persistent biases in the one-factor models. We can also observe in Fig. 5b that the pricing errors of the CSQR, CTOU, and CTOUSV models are almost identical, albeit with slightly smaller oscillations for the CTOUSV model. This feature confirms once again that central tendency is the most relevant extension for pricing futures. Table 4 reports the parameter estimates that we obtain under the real measure in the extensions to the SQR and LOU models we are considering. In the models with central tendency, we observe fast mean-reversion of the VIX to yðtÞ, which in turn mean-reverts rather more slowly to its long-run mean y (i.e., k b k ). Importantly, the estimates of these parameters are very stable in

16 Formally, when we consider a Taylor expansion of the futures price formula of model LOUJ around l=d for l=d 4 0 but small, we only observe deviations from the LOU expression for the second and higher terms. This probably reflects the fact that the conditional mean of the VIX is essentially unaffected by the presence of jumps although they alter skewness and kurtosis.

12 10 8

SQR LOU CTOU

6 4 2 0 −2 −4 −6 −8 −10 Mar06 Oct06

Apr07

Nov07 Jun08 Dec08

Jul09

Jan10 Aug10

12 CTOU CTOUSV CSQR

10 8 6 4 2 0 −2 −4 −6 −8 −10 Mar06 Oct06

Apr07

Nov07 Jun08 Dec08

Jul09

Jan10 Aug10

Fig. 5. Historical evolution of futures pricing errors. (a) Differences between model-based and actual one-month futures prices. (b) Differences between model-based and actual one-month futures prices for the two best performing models. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘CTOUSV’’ introduces central tendency and stochastic volatility in the LOU model. ‘‘CSQR’’ is the concatenated SQR model. One-month actual futures prices have been obtained by interpolation of the prices of the adjacent maturities. Estimation by maximum likelihood for the SQR, LOU and CTOU models, and by pseudo-maximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10.

models with central tendency. In contrast, jump intensities vary substantially depending on the sample period considered for estimation. Specifically, we obtain smaller values of l when we include the crisis period. For the full sample, we estimate around five jumps per year in the LOUJ specification, and almost seven jumps per year in the CTOUJ model. On the other hand, the estimates of the mean-reversion parameter l in the stochastic volatility models tend to be larger when central tendency is not simultaneously included. Since this parameter is also responsible for jump intensity in the OU-G model, this result has two interesting implications. First, a smaller value of l tends to reduce jump activity in Z ðtÞ. Specifically, the expected number of jumps per year decreases from 15

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

379

Table 4 Real measure parameter estimates of the extended models. Estimation by maximum likelihood for the CTOU model, and by pseudo-maximum likelihood in the remaining ones. The data employed includes the VIX index and its options and futures from February 06 until December 10. ‘‘LOUJ’’ introduces jumps in the log-normal Ornstein-Uhlenbeck process (LOU), whose size follows an exponential distribution. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated square root model. Standard errors, displayed in parentheses, have been obtained by using the outer-product of the score to estimate the information matrix. Extension

Estimation

k

k

y

CTOU

Full sample

5.827 (0.017) 6.946 (0.041) 3.632 (0.004) 2.289 (0.008) 0.679 (0.004) 0.573 (0.005) 5.452 (0.011) 6.094 (0.029) 8.903 (0.009) 10.029 (0.038) 3.076 (0.009) 2.575 (0.126)

0.300 (0.003) 0.422 (0.006)

3.019 (0.710) 2.851 (0.867) 3.093 (0.112) 2.897 (0.159) 2.978 (0.002) 2.842 (0.002) 3.021 (0.873) 2.861 (1.055) 2.934 (0.163) 2.826 (0.209) 22.476 (0.645) 19.795 (1.565)

Until Aug08 LOUJ

Full sample Until Aug08

LOUSV

Full sample Until Aug08

CTOUJ

Full sample Until Aug08

CTOUSV

Full sample Until Aug08

CSQR

Full sample Until Aug08

0.285 (0.003) 0.360 (0.006) 0.357 (0.002) 0.477 (0.006) 0.673 (0.021) 0.446 (0.032)

Table 5 Prices of risk in the CTOUSV model. The estimates have been obtained by pseudo-maximum likelihood using data on the VIX index and its options and futures from February 06 until August 10. Standard errors are displayed in parentheses below the estimates, and p-values are reported below the Wald tests. ‘‘CTOUSV’’ introduces central tendency and stochastic volatility in a log-normal Ornstein-Uhlenbeck process. Estimates

Full sample Until Aug 08

Wald tests

Bo

B

Bo ¼ 0

B ¼0

Joint

 1.27 (0.15)  1.19 (0.18)

0.15 (0.26) 0.14 (0.44)

76.35 (0.00) 45.63 (0.00)

0.32 (0.57) 0.11 (0.74)

76.99 (0.00) 46.67 (0.00)

in the LOUSV model to two in the CTOUSV extension. Second, the deviations of oðtÞ from its long-run mean are more persistent in the CTOUSV case because meanreversion is slower the smaller l is, as (18) indicates. These features are also important from a time-series perspective. As mentioned before, central tendency is consistent with ARMA(2,1) dynamics in discrete time, while stochastic volatility introduces GARCH-type persistent variances for the log VIX. 5.3. Risk pricing Table 5 shows the point estimates of the price of risk parameters for our preferred model. We also conduct

l

5.370 (0.109) 14.621 (0.721)

6.671 (0.071) 11.382 (0.222)

d

s

s 0.446 (0.001) 0.447 (0.001)

4.870 (0.037) 7.802 (0.149)

1.037 (0.001) 1.012 (0.002) 0.728 (0.002) 0.382 (0.010)

5.196 (0.013) 5.962 (0.030)

0.836 (0.002) 0.640 (0.004)

6.424 (0.005) 3.732 (0.006)

0.486 (0.001) 0.450 (0.001) 0.222 (0.001) 0.227 (0.003) 2.145 (0.007) 1.360 (0.013)

l

d

14.937 (0.038) 15.631 (0.126)

3.026 (0.007) 3.474 (0.014)

2.140 (0.005) 0.953 (0.013)

0.271 (0.001) 0.170 (0.001)

Wald tests to assess the statistical significance of these estimates. The price of risk related to the VIX Eq. (19), Bo , is highly significant in the two samples. In contrast, B , which is the price of risk related to the innovations in (12), is insignificant in both cases. The negative sign of Bo implies a more adverse distribution in the Q measure, because the VIX mean-reverts towards a higher long-run level than in the P measure. This result, which is consistent with the evidence found by Egloff, Leippold, and Wu (2010), should be taken into account in inferring the market expectation about future values of the VIX from its derivatives. The difference between the properties of the VIX under the real and risk-neutral measures implies, within the framework of our model, that an economically important systematic risk is priced in the market of VIX derivatives. In addition, the statistical significance of the estimate of Bo implies that the impact of the price of risk is larger when the VIX is more volatile (see Eq. (19)). 5.4. Evolution of the factors Fig. 6a shows that the filtered values of the central tendency factor yðtÞ are rather insensitive to the particular specification that we use. We have also found that our preferred CTOUSV model generates almost indistinguishable filtered values for yðtÞ for the two estimation samples that we consider. Not surprisingly, Fig. 6a also confirms that the log of the VIX oscillates around yðtÞ, which in turn changes over time rather more slowly. In fact, the main reason for

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

380

Table 6 Root mean square pricing errors of futures prices by maturity. Estimation of the parameters by maximum likelihood for the SQR, LOU and CTOU models, and by pseudo-maximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10.‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘LOUJ’’ introduces jumps in the LOU model, whose size follows an exponential distribution. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated SQR model. The column labeled N gives the number of prices per category. Maturity Less than 1 month From 1 to 3 months From 3 to 6 months More than 6 months Total

N

SQR

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

CSQR

1,036 2,290 3,093 2,246 8,665

2.607 6.128 9.039 9.370 7.916

1.957 3.953 5.250 6.054 4.892

1.606 2.046 1.561 1.278 1.645

2.068 4.334 5.755 6.489 5.311

2.552 4.332 4.797 4.031 4.262

1.652 2.115 1.626 1.341 1.708

0.982 0.844 0.494 0.567 0.688

1.109 1.029 0.696 0.743 0.862

Fig. 6. Historical evolution of central tendency and stochastic volatility. (a) Filtered yðtÞ for different log-OU models with central tendency. (b) Filtered oðtÞ for different log-OU models with stochastic volatility. (c) One-month volatilities of the VIX. Risk-neutral measure. (d) Logs of one-day-ahead standard deviations of the VIX. Real measure. yðtÞ denotes the time-varying central tendency around which the VIX mean-reverts in central tendency models. oðtÞ denotes the instantaneous volatility of stochastic volatility models. Different vertical scales are used for the VIX and the volatilities in Panel B. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated SQR model. One-month implied vols in Panel C have been obtained by interpolation of the implied vols of options with moneyness 9logðK=Fðt,TÞÞ9 o 0:1, which in turn result from inverting the Black (1976) call price formula. The black line in Panel D is the conditional standard deviation of the VIX, obtained from an ARMA(2,1)-GARCH(1,1) model estimated for the daily data of the log VIX. Only Thursdays are plotted on Panel D to avoid cluttering the picture. Estimation by maximum likelihood for the SQR, LOU and CTOU models, and by pseudomaximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10.

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

central tendency models to work so well during the recent financial crisis is because they allow for large temporal deviations of yðtÞ from its long-term value y , thereby reconciling the large increases of the VIX observed during that period with mean-reversion over the long term. Fig. 6b compares the filtered instantaneous volatilities of the LOUSV and CTOUSV models. Interestingly, both series display an almost identical pattern, with substantial persistent oscillations over time. This figure also shows that large increases in the value of the VIX are associated with volatile periods, as measured by oðtÞ. This is particularly visible in August 2007, October 2008, and May 2010. As with central tendency, we have found that the filtered values of oðtÞ obtained from the CTOUSV model are also quite insensitive to the sample period used to estimate the model parameters.

381

We compute the 30-day-ahead standard deviations of log VðtÞ implied by the CTOUSV model to assess the extent to which the filtered values of oðtÞ make sense. In Fig. 6c we compare those standard deviations with the Black (1976) implied volatilities of at-the-money options that are exactly 30 days from expiration, which we again obtain by interpolation. The high correlation between the two series shows that the filtered values of stochastic volatility are indeed related to the changing perceptions of the market about the standard deviation of the VIX. Finally, Fig. 6d compares the one-day-ahead standard deviations under the real measure implied by the models with the standard deviations obtained from a discrete ARMA(2,1)-GARCH(1,1) model for the log VIX. We only consider the two most relevant extensions to the LOU model to avoid cluttering the picture. As we have already

Fig. 7. Fit of the term structure of futures prices. (a) 21-June-2007. (b) 15-August-2008. (c) 20-November-2008. (d) 31-March-2010. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘LOUJ’’ introduces jumps in the LOU model, whose size follows an exponential distribution. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated SQR model. Estimation by maximum likelihood for the SQR, LOU and CTOU models, and by pseudo-maximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10.

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382

mentioned in Section 4, both the SQR and LOU models yield time-varying variances for the VIX. However, the variance implied by the SQR process seems to be too high until mid2007 and too low afterwards. The LOU model performs better over tranquil periods, but it underestimates actual volatility levels in the most severe phases of the financial crisis. In contrast, the CTOUJ and especially the CTOUSV do a much better job, while the CSQR significantly overestimates the standard deviations during low volatility periods.

5.5. Term structures of derivatives and implied volatility skews So far, we have focused on the overall empirical performance of the different models. In principle, though, our results could change for different time horizons or different degrees of moneyness. For that reason, Table 6 shows the RMSEs of futures contracts for different ranges

of maturity. We observe that the models without central tendency tend to yield larger distortions for longer maturities. In contrast, the models with central tendency are relatively worse at pricing the shortest maturity. Nevertheless, even the worst RMSE of the CTOUSV model is still much smaller than the best RMSE of either the SQR or the LOU models without central tendency. To gain some additional insight, in Fig. 7 we look at the term structure of futures prices for four particularly relevant days. Futures prices were extremely low on June 21, 2007, even though the first warning signals about the impending crisis were starting to appear. Prices had already risen significantly by August 15, 2008, one month before the collapse of Lehman Brothers. Nevertheless, they were much higher on November 20, 2008, which is the day in which the VIX reached its maximum historical closing value at 80.86. The increase is particularly remarkable at the short end of the curve. Since then, though, VIX futures prices significantly came down until the beginning

Table 7 Root mean square pricing errors of call prices by moneyness and maturities. Estimation of the parameters by maximum likelihood for the SQR, LOU and CTOU models, and by pseudo-maximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10. Moneyness is defined as logðK=Fðt,TÞÞ, where K and Fðt,TÞ are the strike and futures prices, respectively. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘LOUJ’’ introduces jumps in the LOU model, whose size follows an exponential distribution. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is a concatenated SQR model. The column labeled N gives the number of prices per category. t denotes time to maturity. Panel A: t o 1 month Moneyness N o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ ½0:1,0:3Þ Z 0:3

2,199 2,080 2,260 2,254 3,085

SQR

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

CSQR

0.155 0.516 0.867 0.771 0.389

0.138 0.373 0.597 0.615 0.361

0.133 0.352 0.527 0.535 0.338

0.142 0.381 0.531 0.414 0.214

0.110 0.140 0.245 0.344 0.253

0.204 0.375 0.509 0.409 0.213

0.142 0.237 0.227 0.187 0.160

0.214 0.601 0.800 0.640 0.349

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

0.250 0.560 0.787 0.864 0.573

0.176 0.381 0.494 0.573 0.448

0.177 0.384 0.486 0.557 0.443

0.174 0.394 0.495 0.452 0.272

0.151 0.166 0.199 0.368 0.328

0.253 0.372 0.476 0.434 0.261

0.214 0.270 0.173 0.192 0.178

SQR

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

0.321 0.486 0.614 0.860 0.758

0.198 0.360 0.451 0.608 0.564

0.215 0.356 0.426 0.583 0.555

0.187 0.370 0.430 0.455 0.312

0.195 0.222 0.254 0.401 0.395

0.251 0.351 0.407 0.432 0.319

0.250 0.280 0.256 0.297 0.289

Panel B: 1 month o t o 3 months Moneyness N SQR o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ , ½0:1,0:3Þ Z 0:3

3,553 4,217 4,701 4,638 8,089

Panel C: t 4 3 months Moneyness N o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ ½0:1,0:3Þ Z 0:3

3,229 3,719 4,491 3,971 5,613

Panel D: All maturities Moneyness N o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ ½0:1,0:3Þ Z 0:3

8,981 10,016 11,452 10,863 16,787

SQR

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

0.260 0.524 0.742 0.844 0.615

0.176 0.372 0.500 0.595 0.476

0.183 0.367 0.472 0.562 0.467

0.171 0.382 0.479 0.446 0.277

0.160 0.185 0.231 0.376 0.340

0.241 0.365 0.457 0.428 0.274

0.213 0.267 0.219 0.235 0.219

CSQR 0.390 0.768 0.792 0.704 0.482

CSQR 0.606 0.806 0.741 0.734 0.648

CSQR 0.451 0.751 0.774 0.702 0.524

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

of April 2010, right before the European sovereign debt crisis. The figure confirms that a central tendency is crucial for the purposes of reproducing the changes in the level and slope of the actual term structure of VIX futures prices. We can also observe that the LOU and LOUJ processes yield futures prices which are almost identical, while the CTOUJ model can be barely distinguished from the CTOU extension. Tables 7 and 8 provide the RMSEs of calls and puts for different ranges of maturity and moneyness. In this case, we generally observe the highest price distortions for atthe-money call and put options. The SQR, CSQR, LOU, LOUJ, CTOU, and CTOUJ models seem to do a better job for moneyness smaller than  0.3. However, once again we can confirm that stochastic volatility models provide the best fit uniformly across all moneyness and maturity ranges. In Figs. 8 and 9 we assess the ability of the different models to fit the average implied volatility skews in Figs. 3b and c. Once again, we only plot the most relevant

383

extensions to the LOU model to avoid cluttering the pictures. In addition to the average implied skews of actual prices, we also consider 5% and 95% percentiles as a measure of dispersion. The figures confirm that the LOU model yields a constant implied volatility for all strikes. In contrast, the SQR and CSQR yield volatility skews but with a negative slope, which is inconsistent with the positive slope in the data. On the other hand, both the CTOUJ and CTOUSV models are able to reproduce this positive slope, with the latter generally providing the best fit. Specifically, the CTOUJ model does not capture the shifts of the implied skews due to rises in volatility and it also performs poorly for low strikes. 6. Conclusions We carry out an extensive empirical analysis of VIX derivatives valuation models. We consider daily prices of futures and European options from February 2006 until December 2010. Therefore, we not only cover an unusually

Table 8 Root mean square pricing errors of put prices by moneyness and maturities. Estimation of the parameters by maximum likelihood for the SQR, LOU and CTOU models, and by pseudo-maximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10. Moneyness is defined as logðK=Fðt,TÞÞ, where K and Fðt,TÞ are the strike and futures prices, respectively. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘LOUJ’’ introduces jumps in the LOU model, whose size follows an exponential distribution. ‘‘CTOU’’ adds central tendency to the LOU process. ‘‘LOUSV’’ denotes a LOU model with stochastic volatility modeled with a Gamma OU Le´vy process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is a concatenated SQR model. The column labeled N gives the number of prices per category. t denotes time to maturity. Panel A: t o 1 month Moneyness N o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ ½0:1,0:3Þ Z 0:3

457 1,657 2,239 1,787 1,385

SQR

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

CSQR

0.268 0.568 0.870 0.833 0.442

0.232 0.415 0.598 0.660 0.409

0.212 0.391 0.533 0.579 0.386

0.248 0.425 0.536 0.464 0.283

0.101 0.141 0.242 0.350 0.275

0.345 0.420 0.517 0.459 0.281

0.199 0.259 0.240 0.192 0.188

0.268 0.636 0.813 0.691 0.400

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

0.304 0.581 0.814 0.968 0.651

0.224 0.399 0.515 0.663 0.512

0.222 0.400 0.507 0.640 0.498

0.231 0.410 0.513 0.510 0.333

0.159 0.160 0.199 0.373 0.346

0.347 0.394 0.496 0.505 0.321

0.244 0.268 0.178 0.202 0.218

0.380 0.746 0.792 0.765 0.538

SQR

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

CSQR

0.366 0.547 0.745 1.068 0.747

0.269 0.411 0.523 0.746 0.566

0.276 0.410 0.515 0.730 0.548

0.279 0.418 0.489 0.556 0.376

0.205 0.240 0.271 0.404 0.337

0.394 0.410 0.476 0.542 0.368

0.260 0.299 0.288 0.344 0.404

0.687 0.810 0.805 0.920 0.628

SQR

LOU

CTOU

LOUJ

LOUSV

CTOUJ

CTOUSV

0.327 0.568 0.812 0.944 0.598

0.245 0.406 0.540 0.679 0.485

0.245 0.402 0.516 0.638 0.468

0.253 0.415 0.513 0.504 0.323

0.175 0.186 0.231 0.371 0.318

0.367 0.404 0.497 0.497 0.315

0.246 0.277 0.228 0.234 0.252

Panel B: 1 month o t o 3 months Moneyness N SQR o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ ½0:1,0:3Þ Z 0:3

1,970 4,274 3,966 2,272 1,505

Panel C: t 43 months Moneyness N o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ ½0:1,0:3Þ Z 0:3

1,698 2,764 2,174 990 633

Panel D: All maturities Moneyness N o 0:3 ½0:3,0:1Þ ½0:1,0:1Þ ½0:1,0:3Þ Z 0:3

4,125 8,695 8,379 5,049 3,523

CSQR

CSQR 0.521 0.748 0.801 0.773 0.507

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0.5 Histogram Data LOU SQR CTOUJ CTOUSV CSQR

0.45 0.4 0.35 0.3

0.45 0.4 0.35 0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1 −0.4

−0.2

0

0.2

0.4

0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.5 0.45

0.45

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1 −0.2

0

0.2

0.4

0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 8. Fit of the implied volatility smirks. (a) Low volatility. 15r t o 45. (b) High volatility. 15 r t o 45. (c) Low volatility. 45r t o 75. (d) High volatility. 45 r t o 75. Short maturities. The lines show the average implied volatility for a certain moneyness and time to maturity (denoted by t) within a given interval. The thick black lines correspond to the average implied volatilities of the actual call prices, while the thin black lines show the 5% and 95% percentiles. Grey bars at the bottom of the plots show the histogram of call prices across maturities. The interval of moneyness in each panel has been chosen to cover the central 90% section of the data. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated SQR model. Moneyness is defined as logðK=Fðt,TÞ. Estimation by maximum likelihood for the SQR, and LOU models, and by pseudomaximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10.

tranquil period, but also the early turbulences that took place between August 2007 and August 2008, the worst months of the recent financial crisis (autumn 2008), as well as the months in 2010 in which the European sovereign debt crisis unfolded. These markedly different periods provide a very useful testing ground to assess the empirical performance of the different pricing models. We estimate the models using not only futures and options data, but also historical data on the VIX itself, which allows us to look at the relationship between real and risk-neutral measures. We initially focus on the two existing meanreversion models: the square root (SQR) and the log-normal Ornstein-Uhlenbeck processes (LOU). Although SQR is more popular in the empirical literature, we find that the LOU model yields a better fit, especially during the crisis. However, both models yield large price distortions during the crisis. In addition, they do not seem to capture either the level or the slope of the term structure of futures prices, or indeed the volatility skews. Part of the problem is that these models implicitly assume that volatility either meanreverts at a simple exponential rate or does not meanrevert at all, which cannot accommodate the long and persistent swings of the VIX observed in our sample. In this sense, we show that the simple AR(1) structure that

they imply in discrete time is not consistent with the empirical evidence of ARMA dynamics for the VIX. In addition, these models are also inconsistent with the strong presence of GARCH-type heteroskedasticity that we find in this volatility index. We investigate the potential sources of mispricing by considering several empirically relevant generalisations. In particular, we consider the concatenated SQR process (CSQR), which substantially extends the SQR model by introducing a time-varying central tendency and allowing for unspanned stochastic volatility. We also extend the LOU model by introducing a time-varying central tendency, jumps, and stochastic volatility. Our parameter estimates indicate that the VIX rapidly mean-reverts to a central tendency, which in turn reverts more slowly to a long-run constant mean. This flexible structure can reconcile the large variations of the VIX over our sample period with mean-reversion to a long-run constant value. Except for the CTOU model, though, it is not generally possible to price derivatives in closed form for the extensions that we consider. For that reason, we obtain the required prices by Fourier inversion of the conditional characteristic function. Interestingly, our results indicate that a time-varying central tendency is crucial for pricing futures, regardless

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

385

0.5 0.45

0.45

0.4

0.4

0.35

0.35

0.3

Histogram Data LOU SQR CTOUJ CTOUSV CSQR

0.25 0.2 0.15 0.1 −0.2

0

0.2

0.4

0.3 0.25 0.2 0.15 0.1

0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.5 0.45

0.45

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1 −0.4

0.1 −0.2

0

0.2

0.4

0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 9. Fit of the implied volatility smirks. (a) Low volatility. 75 r t o 105. (b) High volatility. 75 r t o 105. (c) Low volatility. 105 r t o 135. (d) High volatility. 105 r t o 135. Long maturities. The lines show the average implied volatility for a certain moneyness and time to maturity (denoted by t) within a given interval. The thick black lines correspond to the average implied volatilities of the actual call prices, while the thin black lines show the 5% and 95% percentiles. Grey bars at the bottom of the plots show the histogram of call prices across maturities. The interval of moneyness in each panel has been chosen to cover the central 90% section of the data. ‘‘SQR’’ denotes square root model and ‘‘LOU’’ refers to a log-normal Ornstein-Uhlenbeck process. ‘‘CTOUJ’’ adds central tendency and jumps to the LOU model, while ‘‘CTOUSV’’ introduces central tendency and stochastic volatility. ‘‘CSQR’’ is the concatenated SQR model. Moneyness is defined as logðK=Fðt,TÞ. Estimation by maximum likelihood for the SQR, and LOU models, and by pseudomaximum likelihood in the remaining ones. The data employed in the estimation includes the VIX index and its options and futures from February 06 until December 10.

of whether the model is expressed in levels or logs. We also find evidence of time-varying volatility in the VIX. As expected, stochastic volatility plays a much more important role for options while leaving futures prices almost unaffected. The CSQR model, though, seems unable to generate the positive slope of the option-implied volatility skews. It is also worth mentioning that jumps only provide a minor improvement for options and do not change futures prices (up to first order). Nevertheless, we would like to emphasise that jumps seem to be a relevant feature to describe the historical dynamics of the VIX, even though they only yield second-order gains for pricing VIX derivatives. Importantly, our results remain valid when we focus exclusively on the out-of-sample performance with parameters estimated using data prior to the autumn of 2008. In view of these findings, we conclude that a generalised LOU model that combines a time-varying central tendency with stochastic volatility is needed to obtain a good pricing performance during bull and bear markets, as well as to capture the term structures of VIX futures and options, and the positive slope of the implied volatility skews of options. Interestingly, the price of risk of this specification is highly significant and implies that the VIX mean-reverts towards a higher long-run mean under the risk-neutral measure than

under the real measure. The difference between the properties of the VIX under the real and risk-neutral measures implies that an economically important systematic risk is priced in the market of VIX derivatives. Given the relationship between the observed VIX index and the unobserved integrated volatility of the S&P500, our analysis also has important implications for the models and stationarity of this broad stock index commonly used by participants in markets for stock index options. Specifically, our results imply that stochastic volatility models for the S&P500 should allow for slow mean-reversion by including two volatility factors, and a time-varying volatility of volatility. Amengual (2009) considers such a model for volatility swaps. We could extend our empirical exercise to other recently introduced volatility derivatives such as binary options or American options on VIX futures, or even the futures and options on the CBOE Gold ETF Volatility Index. Sophisticated filtering procedures for the volatility of VIX might also be worth exploring, as well as tractable ways of modeling jump and stochastic volatility risk for Le´vy processes. It would also be interesting to investigate the incremental information content of the contemporaneous observations of the S&P500 over and above the spot VIX

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

386

for pricing VIX futures, and above and beyond VIX futures for pricing VIX options. This question is also relevant for the purposes of integrating the valuation of VIX derivatives with the valuation of the underlying options on the S&P500 that are used to compute this volatility index, as suggested by Lin and Chang (2009). We plan to address these points in subsequent research. Appendix A. Affine model conditional characteristic function The extensions that we consider belong to the class of affine jump-diffusion state processes analysed by Duffie, Pan, and Singleton (2000). In particular, consider an N-dimensional vector YðtÞ that satisfies the diffusion pffiffiffiffiffiffiffiffi dYðtÞ ¼ KðHYðtÞÞ dt þ SðtÞ dWðtÞ þ dZðtÞ, ðA:1Þ where WðtÞ is an N-dimensional vector of independent standard Brownian motions, K is an N  N matrix, H is a vector of dimension N, SðtÞ is a diagonal matrix of dimension N whose ith diagonal element is ci0 þ c0i1 YðtÞ, and finally, ZðtÞ is a multivariate pure jump process with intensity l whose jump amplitudes have joint density f J ðÞ. Duffie, Pan, and Singleton (2000) show that the conditional characteristic function of YðTÞ can be expressed as 0

fY ðt,T,uÞ ¼ E½expðiu YðTÞÞ9IðtÞ ¼ expðj0 ðtÞ þ u0Y ðtÞYðtÞÞ, where j0 ðtÞ and uY ðtÞ satisfy the following system of differential equations:

u_ Y ðtÞ ¼ K0 uY ðtÞ þ 121Y ðtÞ, j_ 0 ðtÞ ¼ H0 KuY ðtÞ þ 12u0Y ðtÞdiagðc0 ÞuY ðtÞ þ l½JðuY ðtÞÞ1,

R where JðuÞ ¼ expðu0 xÞf J ðxÞ dx and 1Y ðtÞ is an N-dimensional vector whose kth element is BY,k ðtÞ ¼ u0Y ðtÞ diagðck1 ÞuY ðtÞ. Appendix B. The CSQR model Given the prices of risk (8) and (9), we can write the CSQR process under the real measure as pffiffiffiffiffiffiffiffiffi P P dVðtÞ ¼ kP ½y ðtÞVðtÞ dt þ s VðtÞ dW v ðtÞ, with P

P

P

dy ðtÞ ¼ k P ½y y ðtÞ dt þ s P

pffiffiffiffiffiffiffiffi yðtÞ dW Py ðtÞ,

where k ¼ kP þ sB, k ¼ k P þ sB ,

y¼

P kP k P y , ðkP þ sBÞðk P þ sB Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s ¼ sP

#

vec½V½XðTÞ9IðtÞ

" ¼

m0

#

v0

" þ

M1

#

V1

XðtÞ,

where m0 , v0 , M1 and V1 are 2  1, 4  1, 2  2 and 4  2 vectors and matrices, respectively, such that " # m0 ¼ ½expðtAÞI6 A1 a, v0 " # " # I2 M1 ¼ expðtAÞ , V1 042 a is a 6  1 vector whose first element is k y and all the other elements are zero; I6 is the identity matrix of order 6; 042 is a 4  2 matrix of zeros; and " # R 024 A¼ , R ðR  I2 Þ þ ðI2  RÞ with

R¼



2 k

0

k

k

 ,

s2

6 6 0

R¼6 6

4 0 0

0

3

7 0 7 7: 0 7 5

s2

It can be tediously shown that m0 þM1 XðtÞ yields (10). Using the results form Appendix A, we can express the conditional characteristic function of this model as

fCSQR ðt,uÞ ¼ E½expðða þ iuÞVðTÞÞ9IðtÞ ¼ exp½jCSQR,0 ðtÞ þ jCSQR, y ðtÞyðtÞ þ jCSQR,V ðtÞVðtÞ, where

j_ CSQR, y ðtÞ ¼ kjCSQR, y ðtÞ þ kjCSQR,V ðtÞ þ 12s 2 j2CSQR, y ðtÞ, j_ CSQR,V ðtÞ ¼ kjCSQR,V ðtÞ þ 12s2 j2CSQR,V ðtÞ, j_ CSQR,0 ðtÞ ¼ k y jCSQR, y ðtÞ, with the conditions jCSQR, y ð0Þ ¼ 0, jCSQR,V ð0Þ ¼ a þ iu, and jCSQR,0 ð0Þ ¼ 0. Furthermore, it can be shown that

jCSQR,V ðtÞ ¼

ða þ iuÞexpðktÞ 2

1ða þ iuÞs

ð1expðktÞÞ 2k

:

Then, we can follow Amengual and Xiu (2012) in showing that the price of a European call option with strike K can be expressed as " # Z expðr tÞ 1 exp½Kða þiuÞ cðt,T,KÞ ¼ Re fCSQR ðt,uÞ du, p ða þiuÞ2 0 where the smoothing parameter a must be such that

ao

2k : s2 ð1expðktÞÞ

kP , k sB Pþ

Appendix C. Extensions of LOU processes

and

yðtÞ ¼

of XðtÞ: " E½XðTÞ9IðtÞ

kP yP ðtÞ: kP þ sB

Let XðtÞ ¼ ½yðtÞ,VðtÞ0 . Following Fackler (2000), it can be shown that the conditional mean and variance of X(T) given information known at time t is the affine function

C.1. General case If we place log VðtÞ as the first element of YðtÞ in (A.1), then the conditional characteristic function of log VðtÞ will be fðt,T,uÞ ¼ fY ðt,T,u0 Þ, where u0 ¼ ðu,0,    ,0Þ0 . Following

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

Carr and Madan (1999), the price of a call option with strike K can then be expressed as Z expða logðKÞÞ 1 expðiu logðKÞÞcðuÞ du, cðt,T,KÞ ¼

p

0

ðC:1Þ where

cðuÞ ¼

expðr tÞfðt,T,uð1þ aÞiÞ a2 þ au2 þ ið1 þ 2aÞu

and a is a smoothing parameter. We evaluate (C.1) by numerical integration. In our experience, a ¼ 1:1 yields good results. Given that the estimation algorithm requires the evaluation of the objective function at many different parameter values, we linearise option prices with respect to oðtÞ in the models with stochastic volatility to speed up the calculations. Our procedure is analogous to the treatment of other stochastic volatility models, which are sometimes linearised to employ the Kalman filter (see, e.g., Trolle and Schwartz, 2009). Specifically, we linearise call prices for day t around the volatility of the previous day as follows:    1 cLOUSV ðt,T,K, oðtÞÞ cLOUSV t,T,K, o t 360  @cLOUSV ðt,T,K,xÞ þ  @x x ¼ oðt1=360Þ    1 :  oðtÞo t 360 Due to the high persistence of oðtÞ, its previous-day value turns out to be a very good predictor, which reduces the approximation error of the above expansion. In fact, the linearisation error, expressed in terms of the RMSEs of options, is very small (below 0.25%). In any case, we calculate the exact pricing errors once we have obtained the final parameter estimates.

By exploiting log-normality, we can write futures prices as F CTOU ðt,T,VðtÞ, k, y, sÞ ¼ expðmCTOU ðt, tÞ þ 0:5j2CTOU ðtÞÞ, while call prices follow the Black (1976) formula with volatility jCTOU ðtÞ. This confirms that the prices of options do not depend on yðtÞ once we condition on the futures price. In terms of time series dynamics, it can be shown that yðtÞ and log VðtÞ jointly follow a Gaussian first-order vector autoregressive process (VAR(1)) if sampled at equally spaced intervals. Specifically, ! ! yðTÞ yðtÞ ¼ gt þ Ft þ et , log VðTÞ log VðtÞ where " gt ¼

and et  iid Nð0, Rt Þ, where Rt is a symmetric 2  2 matrix with elements

s2 ½1expð2ktÞ, 2k   ks 2 1expð2ktÞ 1expððk þ k ÞtÞ Rt ð1,2Þ ¼  , 2k kk kþk Rt ð1,1Þ ¼

and Rt ð2,2Þ ¼ j2CTOU ðtÞ. From here, it is straightforward to obtain the marginal process followed by log VðtÞ, which corresponds to the following ARMA(2,1) model: log VðtÞ ¼ h0 ðtÞ þ h1 ðtÞlog VðttÞ þ h2 ðtÞlog Vðt2tÞ þ uðtÞ þgðtÞuðttÞ, where uðtÞ,uðttÞ,     iid Nð0,p2 ðtÞÞ and h0 ðtÞ ¼ y ð1h1 ðtÞh2 ðtÞÞ,

gðtÞp2 ðtÞ ¼ ðFt ð2,1Þ Ft ð1,1ÞÞRt

  0 , 1

P

P

d log VðtÞ ¼ k½y ðtÞlog VðtÞ dt þ s dW v ðtÞ, P

y,

1expðktÞ kkk ðexpðktÞexpðktÞÞ # expðktÞ 0 Ft ¼ , k kk ½expðktÞexpðktÞ expðktÞ "

h2 ðtÞ ¼ Ft ð1,1ÞFt ð2,2Þ, Given the prices of risk (13) and (14), the diffusions under the real measure can be expressed as P

#

1expðktÞ

h1 ðtÞ ¼ Ft ð1,1Þ þFt ð2,2Þ,

C.2. Central tendency

P

387

P

dy ðtÞ ¼ k ½y y ðtÞ dt þ s dW y ðtÞ,

ð1 þ g 2 ðtÞÞp2 ðtÞ ¼ ðFt ð2,1Þ Ft ð1,1ÞÞRt

Ft ð2,1Þ Ft ð1,1Þ

! þ Rt ð2,2Þ:

where P

y ¼yþ

sB sB þ , k k

and

yP ðtÞ ¼ yðtÞ þ

sB : k

Following Leo´n and Sentana (1997), it can be shown that the conditional distribution of log VðTÞ given information up to time t is Gaussian with mean mCTOU ðt, tÞ given in (15) and variance

j2CTOU ðtÞ ¼

s2 ½1expð2ktÞ 2k 2

3 ktÞ ktÞ  k 2 1expð2 þ 1expð2 2k 2k 2 4 5: þs k þ k ÞtÞ kk 2 1expðð kþk

C.3. Jumps Q

P

Once again, we assume that dW v ðtÞ ¼ dW v ðtÞ þ B dt holds and that jump risk is not priced. Then, it can be shown that log VðtÞ satisfies the diffusion P

P

d log VðtÞ ¼ k½y log VðtÞ dt þ s dW v ðtÞ þ dZðtÞ

l

kd

dt

P

under the real measure, where y ¼ y þ sB=k. The conditional characteristic function reduces to

fLOUJ ðt,T,uÞ ¼ exp½jLOUJ,0 ðtÞ þ jLOUJ,V ðtÞlog VðtÞ, where 

jLOUJ,0 ðtÞ ¼ iu y 



l s2 u2 ½1expð2ktÞ ½1expðktÞ kd 4k

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

388

þ

l

k

log





and

diu expðktÞ diu

KðtÞ ¼

Z t 2

0

and

6 4

jLOUJ,V ðtÞ ¼ iu expðktÞ:

d d iv expðl xÞ þ iuBo ½expðkxÞexpðl xÞ kl

u

½expð2kxÞexpðl xÞ 2ð2kl Þ

7 5

Based on the characteristic function, it is possible to show that

C.4. Stochastic volatility We consider a price of risk such that dW Qv ðtÞ ¼ dW Pv ðtÞ þ p ffiffiffiffiffiffiffiffiffi ffi Bo oðtÞ dt, but we again assume that jump risk related to Z(t) is not priced. Then, the process under the real measure can be expressed as h i pffiffiffiffiffiffiffiffiffiffi B P d log VðtÞ ¼ k y þ o oðtÞlog VðtÞ dt þ oðtÞ dW v ðtÞ,

k

E½log VðTÞ9VðtÞ, oðtÞ ¼ y ½1expðktÞ " # l Bo 1expðktÞ 1expðl tÞ   k d ðkl Þ l þexpðktÞlog VðtÞ E½oðTÞ9VðtÞ, oðtÞ ¼

doðtÞ ¼ l oðtÞ dt þ dZ ðtÞ: Since in this case the processes under the real and riskneutral measures are different, we will describe them separately.

C.4.1. Risk-neutral measure The conditional characteristic function simplifies to " # jLOUSV,0 ðtÞ þ jLOUSV,V ðtÞlog VðtÞ fLOUSV ðt,T,uÞ ¼ exp , þ jLOUSV, o ðtÞoðtÞ

1

d

þ

1 2kl

½1expðl tÞþ expðl tÞoðtÞ,

½expðl tÞexpð2ktÞoðtÞ,

V P ½oðTÞ9VðtÞ, oðtÞ ¼

jLOUSV,V ðtÞ ¼ iu expðktÞ,

Bo ½expðktÞexpðl tÞoðtÞ, kl

V½log VðTÞ9VðtÞ, oðtÞ "  l 1expðl tÞ 1expð2ktÞ ¼  2k ð2kl Þd l  2 2B l 1expð2ktÞ þ 2 o 2 2k d ðkl Þ # 1expð2l tÞ 1expððk þ l ÞtÞ 2 þ kþl 2l

where

1expð2l tÞ

d

2

jLOUSV, o ðtÞ ¼

2

3 dx:

u ½expð2ktÞexpðltÞ, 2ð2klÞ

2

,

and cov½log VðTÞ, oðTÞ9VðtÞ, oðtÞ " # 2Bo l 1expððk þ l ÞtÞ 1expð2l tÞ : ¼ 2  kþl 2l d ðkl Þ

and

jLOUSV,0 ðtÞ ¼ iy u½1expðktÞ þ l½Kðt,uÞt, with Kðt,uÞ ¼

Z t

C.5. Central tendency and jumps

d

u2 0 d 2ð2klÞ½expð2

kxÞexpðlxÞ

dx:

ðC:2Þ

C.4.2. Real measure We can write the characteristic function under the real measure as

For this case, we use the same prices of risk as in the CTOU model and assume that jump risk is not priced. Then, we obtain

P

fPLOUSV ðt,T,uÞ ¼ EP ½expðiu log VðTÞ þ ivoðTÞÞ9VðtÞ, oðtÞ ¼ exp½jPLOUSV,0 ðtÞ þ jPLOUSV,V ðtÞlog VðtÞ þ jPLOUSV, o ðtÞoðtÞ,

jPLOUSV, o ðtÞ ¼ iv expðl tÞiu þ

u2 2ð2kl Þ

Bo ½expðktÞexpðl tÞ kl

½expð2ktÞexpðl tÞ

P

P

under the real measure, where P

y ¼yþ

jPLOUSV,V ðtÞ ¼ iu expðktÞ,

P

l

kd

dy ðtÞ ¼ k ½y y ðtÞ dt þ s dW y ðtÞ,

where

jPLOUSV,0 ðtÞ ¼ iuy ½1expðktÞ þ l ½KðtÞt,

P

P

d log VðtÞ ¼ k½y ðtÞlog VðtÞ dt þ s dW v ðtÞ þ dZðtÞ

sB sB þ k k

and

yP ðtÞ ¼ yðtÞ þ

sB : k

The conditional characteristic function becomes

fCTOUJ ðt,T,uÞ ¼ E½expðiu log VðTÞ þ ivyðTÞ9VðtÞ, yðtÞ ¼ exp½jCTOUJ,0 ðtÞ þ jCTOUJ, y ðtÞyðtÞ þ jCTOUJ,V ðtÞlog VðtÞ,

dt,

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

where

jCTOUSV, y ðtÞ ¼ iu

jCTOUJ,0 ðtÞ ¼ ivy ½1expðktÞ     l kk 1expðktÞ 1expðktÞ  þ iu y  kd kk k k s 2 v2 s2 u2 ½1expð2ktÞ ½1expð2ktÞ  4k 4k 2 3 

s 2 u2  k 2 4 2 kk

1expð2ktÞ ktÞ þ 1expð2 2k 2k 5 k þ k ÞtÞ 2 1expðð kþk

l k ½expðktÞexpðktÞ, kd kk

jCTOUJ, y ðtÞ ¼ iv expðktÞ þ iu

k kk

½expðktÞexpðktÞ,

jCTOUJ,V ðtÞ ¼ iu expðktÞ: Using the characteristic function, we can show that " # 1expðktÞ E½log VðTÞ9VðtÞ, yðtÞ ¼ y k  kk ðexpðktÞexpðktÞÞ

k ½expðktÞexpðktÞyðtÞ þexpðktÞlog VðtÞ, kk

 2  s l V½log VðTÞ9VðtÞ, yðtÞ ¼ þ ½1expð2ktÞ 2k kd2 2 3 ktÞ ktÞ  k 2 1expð2 þ 1expð2 2k 2k 2 4 5, þs k þ k ÞtÞ kk 2 1expðð kþk 2

1expð2ktÞ

C.6. Central tendency and stochastic volatility Q P dW v ðtÞ ¼ dW v ðtÞ þ

Bo oðtÞ dt and dW Qy ðtÞ ¼ dW Py ðtÞ þ B dt, which yields h i pffiffiffiffiffiffiffiffiffiffi B P d log VðtÞ ¼ k yðtÞ þ o oðtÞlog VðtÞ dt þ oðtÞ dW v ðtÞ, k P P dyðtÞ ¼ k ½y yðtÞ dt þ s dW y ðtÞ, doðtÞ ¼ l oðtÞ dt þ dZ ðtÞ, under the real measure, where

Bs y ¼yþ : k P

k

 1expðktÞ



1expðktÞ



k k kk 2 3 1expð2ktÞ 1expð2ktÞ  þ 2 1 k 2k 2k 4 5  s 2 u2 k þ k ÞtÞ 2 kk 2 1expðð kþk

with Kðt,uÞ defined in (C.2). C.6.2. Real measure We can write the characteristic function under the real measure as "

# # iu1 log VðTÞ þ iu2 yðTÞ  ¼ E exp VðtÞ, yðtÞ, oðtÞ  þ iu3 oðTÞ 2 3 P P jCTOUSV,0 ðtÞ þ jCTOUSV,V ðtÞlog VðtÞ 5, ¼ exp4 þ jPCTOUSV, y ðtÞyðtÞ þ jPCTOUSV, o ðtÞoðtÞ "

where

jPCTOUSV,0 ðtÞ ¼ iu2 y ð1expðktÞÞ   kk 1expðktÞ 1expðktÞ þiu1 y  k k kk 2 s 1expð2 kt Þ u22 2k 2 2 3 u21

s 2  k 2 4 2 kk

u1 u2 s 2

jCTOUSV,V ðtÞ ¼ iu expðktÞ, u2 ½expð2ktÞexpðltÞ, 2ð2klÞ

  1expð2ktÞ 1expððk þ k ÞtÞ  2k kþk kk

k

jPCTOUSV,V ðtÞ ¼ iu1 expðktÞ, jPCTOUSV , y ðtÞ ¼ iu2 expðktÞ þ iu1

k ½expðktÞexpðktÞ, kk

jPCTOUSV, o ðtÞ ¼ iu3 expðl tÞiu1 þ

u21 2ð2kl Þ

Bo ½expðktÞexpðl tÞ kl

½expð2ktÞexpðl tÞ,

and KðtÞ ¼

Z t 2 6 4

where

1expð2ktÞ ktÞ þ 1expð2 2k 2k 5 k þ k ÞtÞ 2 1expðð kþk

þ l ½KðtÞt,

0

C.6.1. Risk-neutral measure The conditional characteristic function is " # jCTOUSV,0 ðtÞ þ jCTOUSV,V ðtÞlog VðtÞ , fCTOUSV ðt,T,uÞ ¼ exp þ jCTOUSV, o ðtÞoðtÞ þ jCTOUSV, y ðtÞyðtÞ

jCTOUSV, o ðtÞ ¼

jCTOUSV,0 ðtÞ ¼ iuk y

3

ks 2 4 2k 5 V½log VðTÞ, yðTÞ9VðtÞ, yðtÞ ¼ k þ k ÞtÞ : kk  1expðð kþk

The p ffiffiffiffiffiffiffiffiffiffiprice of risk is such that

and

fP ðt,T,u1 ,u2 ,u3 Þ

and

þ

k ½expðktÞexpðktÞ, kk

þ l½Kðt,uÞt,

  1expð2ktÞ 1expððk þ k ÞtÞ uvs 2  2k kþk kk   l diu expðktÞ þ log k diu

k

iu

389

d diu3 expðl xÞ þ iu1 Bo ½expðkxÞexpðl xÞ kl

u21 ½expð2kxÞexpðl xÞ



3 dx: 7 5

2ð2kl Þ

Based on the characteristic function, it is possible to show that E½log VðTÞ9VðtÞ, yðtÞ, oðtÞ   P kk 1expðktÞ 1expðktÞ ¼y  k k kk " # l Bo 1expðktÞ 1expðl tÞ   k d ðkl Þ l þexpðktÞlog VðtÞ

J. Mencı´a, E. Sentana / Journal of Financial Economics 108 (2013) 367–391

390

k ½expðktÞexpðktÞyðtÞ kk B  o ½expðktÞexpðl tÞoðtÞ, kl

and

þ

h 2 i exp 2sk expðktÞ 1 h 2i : corr½VðTÞ,VðtÞ ¼ exp 2sk 1

P

E½yðTÞ9VðtÞ, yðtÞ, oðtÞ ¼ y ½1expðktÞ þexpðktÞyðtÞ, E½oðTÞ9VðtÞ, yðtÞ, oðtÞ ¼

1

d

½1expðl tÞ þ expðl tÞoðtÞ,

References

V½log VðTÞ9VðtÞ, yðtÞ, oðtÞ " # l 1expðl tÞ 1expð2ktÞ  ¼ 2k ð2kl Þd l  k 2 1expð2ktÞ 1expð2ktÞ þ þ s2 2k 2k kk  1expððk þ k ÞtÞ 2 kþk " 2B2o l 1expð2ktÞ 1expð2l tÞ þ þ 2 2 2k 2l d ðkl Þ # 1expððk þ l ÞtÞ 2 kþl þ

1 2kl

½expðl tÞexpð2ktÞoðtÞ,

V½yðTÞ9VðtÞ, yðtÞ, oðtÞ ¼ s 2 V½oðTÞ9VðtÞ, yðtÞ, oðtÞ ¼

1expð2ktÞ , 2k

1expð2l tÞ

d

2

, 2

1expð2ktÞ

3

s2k 4 2k 5 cov½log VðTÞ, yðTÞ9VðtÞ, yðtÞ, oðtÞ ¼ k þ k ÞtÞ , kk  1expðð kþk and 2

1expððk þ l ÞtÞ

3

2B l 6 kþl 7 cov½log VðTÞ, oðTÞ9VðtÞ, yðtÞ, oðtÞ ¼  2 o 4 5: l tÞ d ðkl Þ  1expð2 2l

Appendix D. One-factor model variances and autocorrelations Let V(t) follow the SQR process. Then, it can be shown that V½VðTÞ9VðtÞ ¼

s2 ð1expðktÞÞ2 2k s2 þ expðktÞð1expðktÞÞVðtÞ, k

and corr½VðTÞ,VðtÞ ¼ expðktÞ. Alternatively, if V(t) follows the LOU process, it holds that   s2 V½VðTÞ9VðtÞ ¼ exp 2y ð1expðktÞÞ þ ð1expð2ktÞÞ 2k   2   s ð1expð2ktÞÞ 1 ½VðtÞ2 expðktÞ  exp 2k

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